Proceedigs of the 5th WSEAS It. Cof. o SIMULATION, MODELING AND OPTIMIZATION, Corfu, Greece, August 7-9, 005 (pp488-49 Realized volatility estimatio: ew simulatio approach ad empirical study results JULIA BONDARENKO, NIKOS MASTORAKIS Istitute of Applied System Aalysis Natioal Techical Uiversity of Ukraie "Kyiv Politechic Istitute" (NTUU KPI Peremogy Prospect 37 056 Kyiv UKRAINE World Scietific ad Egieerig Academy ad Society (WSEAS Ag. Ioaou Theologou 7-3 5773 Zographou, Athes GREECE http://www.wseas.com/mastorakis/idex.html Abstract: - The paper empirically ivestigates several daily volatility estimators for the DAX idex. Realized volatility is computed by meas of stadard, Parkiso, Garma-Klass estimators, which use the daily data samples, ad also Aderse estimator based o the itraday iformatio observed over time itervals of differet sizes. A Mote Carlo simulatio is coducted for two cases of uderlyig security fluctuatio the diffusio process ad the process based o the telegrapher process; the theoretical results are compared with volatility values obtaied from the studied estimators. Key-Words: - realized volatility, Mote Carlo simulatio, rage estimators, geometric Browia motio, telegrapher (Kac process, estimator s efficiecy. Itroductio I geeral, volatility is defied as fluctuatios i the value of ay fiacial security or i a portfolio of securities, ad cosidered as a measure of market risk. The traditioal method for modelig of uderlyig security prices assumes a radom walk described by a geometric Browia motio process []. Give the curret price S ( 0 of a uderlyig, the future uderlyig price S ( t follows the stochastic differetial equatio: d S( t = µ S( t dt + σ S( t dw ( t, ( where W ( t is a stadard Browia motio, µ represets expected rate of chage or drift rate of the process, ad σ represets volatility, 0 t T. Usig equatio (, the price of a Europea call V ( S, t with a strike price K ad maturity date T ca be foud by meas the Black-Scholes formula [] as follows r T t V ( S, t = S Φ d K e d, ( ( ( ( d ( S / K + ( r+ σ ( T t l =, σ T t where r is a risk free rate, ( x d ( S / K + ( r σ ( T t l =, σ T t Φ is the cumulative ormal probability for a stadard ormal radom variable. The value of σ ca be techically expressed i several ways. I the valuatio of optios, the meaigs of implied volatility ad historical volatility are the most used. Implied volatility is estimated from traded optio prices. Puttig the curret market price of a optio V ~ ito (, we ca compute the value of implied volatility as a uique ~ solutio of the equatio V = V ( σ. Historical (realized volatility describes volatility observed i a security over a give period of time. Price movemets i the security (historical data are recorded at fixed time itervals over a give period. As before, it is assumed that prices are logormally distributed. Usig recet historical data provides better iformatio o the curret level of volatility. There is much discussio over the best method of calculatig the historic volatility. The most usual
Proceedigs of the 5th WSEAS It. Cof. o SIMULATION, MODELING AND OPTIMIZATION, Corfu, Greece, August 7-9, 005 (pp488-49 ad traditioal measure is a stadard deviatio of the log-differeced close prices of asset. Other measures, such as Parkiso s extreme value estimator [3], Garma ad Klass rage-based estimator [4], Aderse s itraday estimator [5], improve the efficiecy of realized volatility measures by usig iformatio embedded i daily high, low, ope prices ad high-frequecy itraday data. The remaider of the paper is orgaized as follows. I Sectio, we start off with the features of data. Further, the sectio presets the realized volatility estimators, which will be employed, ad also describes a Mote Carlo simulatio techique for two differet types of uderlyig process. Sectio 3 presets the simulatio. Sectio 4 reports the empirical results. Coclusios are draw ad suggestios for future research offered i sectio 4. Problem Formulatio.. Data The data set we have used cosists of daily prices for DAX idex from Jauary 8, 996 to Jauary 8, 997. There are all together 5 tradig days. The average daily retur for this period is 0.0008954. Itraday returs are obtaied by samplig from the iitial grid of oe-miute prices o Jauary 8, 997, from 8:30 a.m. to 7:05 p.m. GMT+, ad also calculated at 5 miute, 5 miute ad 30 miute retur period. To aualize the realized volatility for ay give day, we have to multiply it by the square root of the umber of tradig days i a year... Realized volatility estimators The classical volatility estimator is defied as a stadard deviatio of the daily close to close returs: σ cc = ( rt r, (3 Ct where r t = l is a price retur, C t is a close C t price o tradig day t, t = :, r = r t. Parkiso estimator [3] is based o the daily log price rage, which is defied as the differece betwee the daily high ad low log-price, that is: σ p = k p t, (4 H t where pt = l, L t ad Lt the highest ad lowest prices o day t, H t are respectively t = :, k = 0.60. l It was prove that the rage estimator of daily volatility (4 is approximately five times more efficiet tha the estimator based o squared daily close price returs (3. Usig also additioal iformatio embedded i daily ope prices, Garma ad Klass [4] have improved efficiecy ad suggested the followig volatility estimator: σ gk = pt ( l qt, (5 Ct where qt = l, O t is a ope price o day t, Ot t = :. The estimator (5 is more tha seve times more efficiet tha (3. Aderse [5] has show that the accuracy of volatility estimatio ca be also more improved by exploitig high frequecy data. The proposed estimator ( itegrated volatility was costructed as the sum of squared itraday returs, m it, t = R t, j j= σ, (6 where σ is a realized volatility o day t, it, t R t, j is a squared itraday retur, ad m is a umber of samples per day. To avoid a bias problem, it s reasoable to filtrate tick-by-tick price series ad take a time iterval of 5-5 miutes..3. Mote Carlo simulatio The uderlyig asset price process is assumed to follow a oe-dimesioal diffusio process (. Applyig the Itô s Lemma, we ca write the logarithm of the process as σ d l S( t = µ dt + σ dw ( t. (7 A discrete approximatio for (7 has a form σ i l S( ti l S( ti = µ + M (8 + σ ( W ( t W ( t, i i i
Proceedigs of the 5th WSEAS It. Cof. o SIMULATION, MODELING AND OPTIMIZATION, Corfu, Greece, August 7-9, 005 (pp488-49 3 where M is a umber of time itervals, i = : M, W ( t i W ( t i ~ 0,. I the Mote Carlo M procedure [6], we employ the equatio (8, with M = 000. A Mote Carlo simulatio with 600 produced paths gives us the a bechmark. The aual volatility is chose to be a costat σ i = σ, equal 6 %, ad close to the actual volatility for DAX idex durig the observed period. Usig aual retur value, the drift coefficiet ca be calculated σ via µ = 0.38 + = 0. 366..4. Telegrapher process This model of stochastic evolutio was cosidered first i [7]. A described process ca be viewed as followig: a poit is ruig o the real lie with a costat velocity v ; the poit s motio is cotrolled by the Poisso process N ( t with parameter λ. That meas that the poit starts to move from the origi i oe ad the chages istataeously the directio at the momet of the Poisso evet comig; moves durig a radom time iterval util the ext Poisso evet, ad so o. The related process N ( ( ( s t v ξ dt (9 = t 0 givig the positio of the poit, is called telegrapher process, or Kac process. This mathematical model was studied also i [8, 9]. Furthermore, a applicatio of the cosidered stochastic process modificatio i mathematical fiace was suggested i [0]. So, there are discrete, radom poits i the time where the chages occur, what is close to the iterpretig market price movemets. Let s ote also, that if a frequecy of poit s directio chages is high ( λ, v v, ad c = cost, c > 0, the process λ ξ ( t asymptotically is a Browia motio. A approach based o the Mote Carlo simulatio described i the previous sectio, is coducted for the process (9 as well. Discrete approximatio of the correspodig process provides, aalogously to (8, a formula σ l S( ti l S( ti = µ + M (0 + σ ( ξ ( t i ξ ( t i, where as before σ = 0. 3, µ = 0. 505, ad we ra a simulatio procedure also 600 times. The parameters λ ad v are take to be equal 50 ad 5 correspodigly. The further results compariso with other volatility estimators is carried out. 3 Empirical aalysis First of all, i Table we preset a descriptive statistics of the returs for historical data, both daily (close-close prices ad itraday (itervals, 5, 5, 30 ad 60 miutes. Table. Statistics of returs Value Daily mi Mea St. dev. Max Mi Skewess 8.95E- 8.E-.3E- 0-4.4E- 0-9.08E- 0 7.5E- 06.83E-.E- -.E- -4.E- 0 Itraday, with iterval 5 5 30 mi mi mi 3.63E- 05 4.E-.59E- -9.98E- 4.5E- 0.6E- 6.76E-.6E- -.87E- -3.8E- 0.3E-.0E-.4E- -.35E- -5.47E- 0 60 mi.5e-.68e-.48e- -3.9E- -7.64E- 0 Kurtosis 6.557 9.730 4.754 3.797 3.733.990 Mea returs of the idex icrease as iterval duratio becomes loger. The mea estimates for itraday data provide a reasoable proxy for daily data: the daily mea is approximately four times greater tha the itraday meas for itervals 30 ad 60 mi, eight times greater tha the itraday mea for iterval 5 mi, twety five times greater tha the itraday mea for iterval 5 mi, ad hudred twety three times greater tha the itraday mea for iterval mi. There is the same tedecy for differet time itervals stadard deviatio: the estimate for 60 mi iterval is about % of the daily stadard deviatio, for itervals 30, 5, 5 ad mi these magitudes are 3%, 8.4%, 5% ad.3% correspodigly. Maximal ad miimal daily magitudes are by a order greater tha itraday oes. Returs, as a rule, are slightly egatively skewed, ad for the case of 5 mi iterval are ot far from ormality. The itraday returs take with 5 mi iterval have a positive skewess. Kurtosis values show, that returs i geeral are fat-tailed, but for itraday data they declie as a legth of time
Proceedigs of the 5th WSEAS It. Cof. o SIMULATION, MODELING AND OPTIMIZATION, Corfu, Greece, August 7-9, 005 (pp488-49 4 iterval icreases. For 5, 30 ad 60 mi the value of kurtosis is close to the ormal case. The kurtosis for daily data is twice greater tha the ormal oe. Table reports daily volatility estimates derived from classical, Parkiso ad Garma-Klass estimators, ad also from simulatio results for the cases, whe uderlyig security is modelled i frameworks of both stadard Browia motio ad telegrapher (Kac process. As a compariso criterio, we use a efficiecy of each estimator. I the capacity of bechmark we take daily volatility obtaied from the stadard deviatio approach (close-close prices estimator. Volatility values obtaied from Mote Carlo simulatio procedures for diffusio ad Kac processes ( ad (9 are calculated as a mea from 600 realizatios. The efficiecy of arbitrary estimator is defied the by the ratio of the variace of kow estimator to the variace of arbitrary estimator: Var ( ( CC Estimator Eff estimator =. Var( estimator The magitude Eff larger tha meas the variace of the cosidered estimator is smaller tha the variace of the bechmark oe. Hece, the ivestigated estimator is more efficiet. Table. Daily volatility estimates. Simulatio results ad rage estimators. Value Daily Estimator volatility Efficiecy MC-simulatio (GBM 0.005060.556748 MC-simulatio (Kac process 0.05555 3.639743 Classical 0.0080879 Parkiso 0.0847 5.98636966 Garma-Klass 0.0098 6.7589560 The efficiecy of the bechmark estimator is supposed to be equal. Comparig with theoretical value based o Mote Carlo simulatio for geometric Browia motio, the efficiecy is about,5 times smaller. The higher efficiecy (more tha 3 times is achieved by applicatio of the telegrapher (Kac process istead of the stadard Browia motio. Icludig high/low prices i estimator provides a estimator, which is almost 6 times more efficiet (Parkiso estimator, ad additioal use of ope/close prices a estimator, which is 6, 7 times more efficiet (Garma-Klass estimator, tha the bechmark. I geeral, the rage-based estimates are dowward biased. It s show also via the Mote Carlo study: value of Parkiso estimator is oly 65 % of Mote Carlo estimate for stadard Browia motio ad 7 % for Kac process. I the case of Garma-Klass estimator these values equal 6 % ad 69 % correspodigly. Volatility estimates based o itraday iformatio, are preseted i Table 3. This approach defiitely improves a efficiecy of estimatio relative to the classical oe. Table 3. Daily volatility estimates. Itraday prices based (Aderse estimator. Value Daily Iterval volatility Efficiecy miute 0.0445 3.757698 5 miutes 0.00793 3.8858543 5 miutes 0.08849 4.764373 30 miutes 0.054 3.95897895 60 miutes 0.09757 3.65854 Squared itraday returs are biased upwards, ad the bias value is larger whe the data are sampled more frequetly ( ad 5 mi itervals. The highest efficiecy is attaied i the case of 5 mi iterval, it s about 4, 3 times larger tha for the classical close-close price estimator. The 30 ad 60 mi itervals estimates still remai less biased because of smaller umber of data, but have the greater variaces, which make the efficiecy vaishig. The estimate magitude for 5 mi iterval makes 77 % of Mote Carlo estimate for stadard Browia motio ad 87 % for telegrapher process. 4 Coclusio I this study, we cosidered several empirical approaches to the realized volatility measure for the DAX idex prices. The several estimatio methods are employed classical close-close price estimator, rage estimators (Parkiso ad Garma-Klass ad Aderse estimator. The empirical results show, that the rage-based estimators are highly efficiet, but are dowward-biased, that was established through a Mote Carlo study. Our Mote Carlo results also demostrate that by usig the alterative telegrapher process istead of stadard Browia motio, we ca improve estimatio efficiecy. It is cocluded that usig the highest available frequecy of itraday data ( ad 5 miutes i our case leads to upward-biased daily volatility estimates; effect of superior estimatig takes place also for the 30 ad 60 mi itervals due to the larger variace values. A obvious directio for future research would be a further theoretical ad empirical study of the telegrapher process ad its modificatios, aalysis
Proceedigs of the 5th WSEAS It. Cof. o SIMULATION, MODELING AND OPTIMIZATION, Corfu, Greece, August 7-9, 005 (pp488-49 ad compariso with various types of estimators, ad the developmet of volatility forecastig techiques o the basis of proposed process with takig ito accout the market microstructures for cocrete fiacial securities. Refereces: [] P. A. Samuelso, Ratioal Theory of Warrat Pricig, Idustrial Maagemet Review, Vol. 6, 965, pp. 3-3. [] Black F. ad Scholes M.S. (973: The pricig of optios ad corporate liabilities. Joural of Political Ecoomy 8, pp. 637-659. [3] M. Parkiso, The extreme value method for estimatig the variace of the rate of retur. Joural of Busiess, Vol. 53, 980, pp. 6-68. [4] M. Garma ad M. Klass, O the estimatio of security price volatilities from historical data, Joural of Busiess, Vol. 53, 980, pp. 67-78. [5] T. G. Aderse, Some reflectios o aalysis of high-frequecy data, Joural of Busiess & Ecoomic Statistics, Vol. 8, 000, pp. 46-53. [6] P. Glasserma, Mote Carlo methods i fiacial egieerig, Spriger Verlag, 0, 596 p. [7] M. Kac, A Stochastic Model Related to the Telegrapher.s Equatio. Rocky Moutai Joural Math., Vol. 4, No. 3, 974, pp. 497-509. [8] E. Orsigher, A plaar radom motio govered by the two-dimesioal telegraph equatio, Joural of Applied Probability, Vol., 986, pp. 385-397. [9] A. D. Kolesik ad A. F. Turbi, A.F., Symmetrical radom evolutios i R, Doklad. Akadem. Nauk Ukraie, Vol., 990, pp. -4 (i russia. [0] J. Bodareko ad N. Brager, A alterative model of fiacial assets prices dyamics, accepted by Computig ad Visualizatio i Sciece, 0. 5